5.3 The definite integral 141
In general, an arbitrary function is neither even nor odd;f(−x) 1 ≠ 1 ±f(x). It is always
possible however to express a function as the sum of an even component and an odd
component; we can write
= 1 f
+1 (x) 1 + 1 f
−1 (x)
(5.22)
wheref
+(x)is the even component off(x)andf_(x)is the odd component. For
example, an arbitrary polynomial
f(x) 1 = 1 a
01 + 1 a
1x 1 + 1 a
2x
21 +1-1+ 1 a
nx
ncan be written as
and, because every even power of xis an even function whilst every odd power of xis
an odd function, the first set of terms forms the even component of the polynomial
and the second set forms the odd component.
The integral properties of functions of well-defined symmetry are of great
importance in the physical sciences. The area represented by the integral
(5.23)
is the sum of the areas to the left and right of thex 1 = 10 axis:
(5.24)
Iff(x)is an even function of x, the two areasA
<andA
>are equal in magnitude
and sign, and the total area is twice each of them:
(5.25)
On the other hand, iff(x)is an odd function of x, thenA
<and A
>are equal in
magnitude but have oppositesigns: A
<1 = 1 −A
>, and the value of the integral is zero:
(5.26)
In the general case, whenf(x)has no particular symmetry, the integral is equal to the
integral of the even component:
(5.27)
ZZ Z Z
−+−++−+−+=+=
aaaaaaaf x dx() f x dx() f x dx() 2
0ffxdx
+()
Z
−+=
aafxdx() 0 if fx()odd
ZZ
−++=,
aaafxdx() 2 fxdx() fx()
0if even
A fxdx fxdx A A
aa=+=+
−+<>ZZ
0() ()
0Afxdx
aa=
−+Z ()
fx a ax ax()=+++ ax ax ax
++++
0224413355
fx()=+−fx f x() ( ) fx f x() ( )
+−−
1
2
1
2